Tons of heroes face trials in their adventures: think of all of the challenges and monsters Percy Jackson faces, for example. The coconut tree is a trial Phillip must face. He's afraid to climb the tree because he's blind, but he does so for the hungry Timothy's sake.
Squatting near me, his teeth crunching the coconut, Timothy said, "You see, Phill-eep, you do not need d'eye now. You 'ave done widout d'eye what I couldn't do wid my whole body."
It was almost as if I'd graduated from the survival course that Timothy had been putting me through since we landed on the cay. (13.49-50)
Phillip's climbing the tree tells us that he has reached a certain level of maturation. He is acting as a provider for Timothy now; he has conquered his fears and can survive on his own.